Linking number of grid models

TL;DR

This paper investigates the statistical properties of linking numbers in random grid models, revealing asymptotic behaviors and distribution convergence as the grid size grows large.

Contribution

It introduces a detailed analysis of the moments and distribution of linking numbers in random grid links, including asymptotic growth and convergence results.

Findings

The $u$th moment of the linking number is a polynomial in grid size with degree at most $u$.

All odd moments of the linking number vanish.

The distribution of the normalized linking number converges weakly as grid size increases.

Abstract

This paper studies the linking numbers of random links within the grid model. The linking number is treated as a random variable on the isotopy classes of 2-component links, with the paper exploring its asymptotic growth as the diagram size increases. The main result is that the $u$th moment of the linking number for a random link is a polynomial in the grid size with degree $d\leq u$, and all odd moments vanishing. The limits of the moments of the normalized linking number are computed, and it is shown that the distribution of the normalized linking number converges weakly as the grid size tends to infinity.